in which the \(Z^{\prime }\)  decay mode into DM pair \(\chi ^{\star } \, \chi \)  is active for the condition \(m_{Z^{\prime }} > 2 \, m_{\chi }\) .
The relic density is plotted as function of the DM scalar mass \(m_{\chi }\)  in the figure (REF ) for \(m_{Z^{\prime }}=5.36\)  TeV
and the \(U(1)_{X}\)  coupling constant \(g_{X}=0.36\)  constraints by the parametrization in (REF ).
The DM mass is assumed at the range \(0.5\)  TeV to \(5.5\)  TeV. The dashed line sets the actual DM observed relic density at \(\Omega _{DM}h^2=0.12 \, \pm \, 0.0012\)  [1]}. The intersection of DM abundance with the black curve yields the two solutions for DM masses : \(m_{\chi }=1.701\)  TeV and \(m_{\chi }=4.248\)  TeV. If \(Z^{\prime }\)  decays into DM pair \(\chi ^{\ast } \, \chi \) , the first solution at \(m_{\chi }=1.701\)  TeV is kinetically allowed for the \(Z^{\prime }\)  mass of \(5.36\)  TeV. The minimum peak in the relic density curve is at \(m_{\chi }=2.70\)  TeV, that is approximately \(m_{Z^{\prime }} \approx 2 \, m_{\chi }\) .
<FIGURE>